Module 1 · Data and returns Core

Classify financial data and calculate simple and log returns correctly.

Kinds of data

• Time series: one item observed over time, such as a stock's monthly closing price for ten years.
• Cross-section: many items observed at one time, such as the P/E ratios of 500 companies today.
• Panel: many items over time.

Data are also numerical (continuous like returns, or discrete like the number of defaults) or categorical (sector, rating).

Population and sample

A population is the full set of interest; a sample is the part observed. Every return history is a sample from the unknown process that generates returns. Statistics is the craft of reasoning from the sample to that process, with an honest statement of the uncertainty.

Prices are not the raw material; returns are

Prices trend and their scale differs by stock, so statistics are computed on returns.

• Simple return = (P1 minus P0 + dividend) / P0.
• Log (continuously compounded) return = ln(P1 / P0).

Which to use

• Simple returns add across assets: a portfolio's return is the weighted average of its holdings' simple returns.
• Log returns add across time: the log return over two periods is the sum of the two log returns. They are also closer to symmetric, which suits statistical models.
• For small returns the two are almost equal. Report results to people in simple returns.

Getting clean data

• Use prices adjusted for splits and dividends (total return), or returns will show false crashes on split dates and understate income-paying assets.
• Match frequency to purpose: daily for risk measurement, monthly for performance and beta.
• Check for missing days, stale prices in thinly traded securities and obvious errors.
• Keep currency consistent.
• Record the source and download date.

How much data

More observations reduce estimation error, but old data may describe a different world. Five years of monthly returns (60 points) is a common compromise for beta; several years of daily data for volatility.

More worked examples

Excess returns

Many tools in this course use returns above the risk-free rate. A stock that returned 0.8% in a month when treasury bills paid 0.3% earned an excess return of 0.5%. Excess returns measure what the investor earned for taking risk.

Annualizing log returns

The annual log return is the sum of the twelve monthly log returns, and the mean monthly log return times 12 is the annualized mean. A sum of 0.06 converts back to a simple annual return of e^0.06 minus 1 = 6.18%.

Timing traps in data

Markets close at different times. Daily returns of a Japanese stock and the S&P 500 are measured over different hours, so news that hits Tokyo's next session shows up a day later; daily correlations between them look lower than they are. Holidays that close one market but not another cause the same problem. Weekly or monthly data reduce the issue.

Corporate actions

Spin-offs, special dividends, rights issues and mergers all create jumps in unadjusted prices. Data providers adjust for most of them, but not always consistently. When a return series shows a single extreme value, check for a corporate action before treating it as a market move.

Put it into practice